How Do You Calculate Acceleration in a Two-Mass Pulley System?

[tascja]

Homework Statement

Two objects of masses M and m are attached to one another with a string and pulley. Object M sits on the horizontal surface, which allows for frictionless motion. The second object m is hanging over the edge of the table. The string tied to the two objects is massless and passes over a massless pulley that rotates without friction. If M=4.53 kg and m=1.89 kg, what is the magnitude of the acceleration of the sliding object?

Homework Equations

Fnet = ma

The Attempt at a Solution

I know that the tension will be uniform across the entire length of the string. But I am not sure how to write out the equations for everything. I am also confused about the one object moving in the y direction while the other moves in the x direction.. An explanation about how to set up the equations would be greatly appreciated!

 

[LowlyPion]

What force is acting on the horizontally movable block?

And what forces are acting on the vertically mobile block.

Draw a diagram of the 2 in isolation and recall that the Tension in both pictures must be the same.

 

[tascja]

So for the block sitting on the table it has the following forces acting on it:
Normal force
Tension
Gravity
* the normal force and gravity must equal 0 because it is not moving in the y direction, thus Fnet = T
For the object that is in the air, the following forces act on it:
Tension
Gravity
* no forces in the x direction, thus Fnet = T - mg

Newtons 2 law: Fnet = ma
so do i just isolate T in one of the equations to plug into the other?
and do i have to add both masses together when looking at the hanging block?

 

[LowlyPion]

So for the block sitting on the table it has the following forces acting on it:
Normal force
Tension
Gravity
* the normal force and gravity must equal 0 because it is not moving in the y direction, thus Fnet = T
For the object that is in the air, the following forces act on it:
Tension
Gravity
* no forces in the x direction, thus Fnet = T - mg

Newtons 2 law: Fnet = ma

Good.

so do i just isolate T in one of the equations to plug into the other?
and do i have to add both masses together when looking at the hanging block?

What is the acceleration of the system then?

What is the force acting on both masses.

 

[tascja]

What is the acceleration of the system then?

so gravity is the only force acting as an acceleration to the system...?

What is the force acting on both masses.

tension...?

 

[xxChrisxx]

Why would you think you would add the masses together for the hanging block?

 

[tascja]

because isn't it like the hanging object is `pulling` the object on the table? ...

 

[LowlyPion]

so gravity is the only force acting as an acceleration to the system...?

tension...?

The force of the hanging block is your only force on the system. Mv*g = (Mv + Mh)*a where Mv is vertical and Mh is horizontal.

Your tension is then Mh*a, but it is internal to the system.

 

[tascja]

ok so to get that formula: Mv*g = (Mv + Mh)*a
did you just add the Fnet of each object together?

 

[LowlyPion]

ok so to get that formula: Mv*g = (Mv + Mh)*a
did you just add the Fnet of each object together?

Not really. I just looked at it.

∑ Ext Forces = ∑ mass * acceleration

It's true for this case anyway.

 

[tascja]

Not really. I just looked at it.

∑ Ext Forces = ∑ mass * acceleration

It's true for this case anyway.

Will that generally be true for systems where there are 2 or more masses attached by a string?

 

[xxChrisxx]

Ok this has got me confused now. Whats all this adding masses got to do with anything.

You want the acceleration for block M, F=ma for it. You find the T=mg from the other FBD. Plug T=mg into your FBD for the block M and get out a.

EDIT: If I am missing something here, be kind and don't make me look all silly.

 

[tascja]

well i think it is because both blocks will have the same acceleration.. and the only force acting to accelerate the 2 blocks is Fg for the hanging block. and i believe because this force is `pulling` both of the objects, both masses need to be added together when you find it...

 

[tascja]

im not sure if that really makes sense.. lol

 

[xxChrisxx]

They won't have the same acceleration as they have different masses. Fg is only accelerating the hanging block, this then provides a force (T) to the stationary block.

The frictionless pully allows you to draw two separate free body diagrams with T being the same in both.

 

[LowlyPion]

Will that generally be true for systems where there are 2 or more masses attached by a string?

No. It's true here because the two masses are effectively joined as one without any other complicating factors, like the other mass having other forces acting on it like a ramp or a double pulley or any other hellish invention like an anchored or free-acting pulley that creates mechanical advantages. So no it is not a general equation.

 

[LowlyPion]

They won't have the same acceleration as they have different masses. Fg is only accelerating the hanging block, this then provides a force (T) to the stationary block.

The frictionless pully allows you to draw two separate free body diagrams with T being the same in both.

Actually you do have the same acceleration. You must. The string doesn't change length. In this case then the weight of the hanging block is the motive force to be accelerating both blocks - together as one.

Making separate drawings is the correct method. I took a shortcut that I can see is not necessarily useful for understanding.

 

[xxChrisxx]

Im not convinced.

Edit: god knows how a statics problem can have me confused. must be time for bed.

 

[LowlyPion]

Im not convinced.

Work it out then. T eliminates readily enough.

 

[xxChrisxx]

I forgot about the system acceleration, knew there was something utterly wrong with the way I was thinking. Looks like its time for me to refresh basic statics.

Definately time for bed then.

 

[LowlyPion]

Good. Let's put the problem to bed then too.